So what happens if the government prints 10 times the number of paper dollars without a corresponding increase in the amount of silver-containing coins? The same thing that happened in 1965 - the silver-containing coins get yanked out of circulation by ordinary folks as the price of silver rises to the point where the silver content is worth more than the face value of the coin.
I think that what you are saying is that the purchasing power of the silver-containing coin will remain constant - and there I basically agree with you (unless the supply of silver suddenly jumps or declines). However, without the supply of paper currency somehow limited such that the "reserve ratio" of silver to paper remains roughly constant, then the paper dollars will lose value compared to the coins containing silver.
The premis behind the silver alloy dollar coin is that the mint will continue to offer them one-for-one for paper bills. Once the mint stops that practice the situation reverts to Gresham’s law. As long as the mint continues that practice, our dollar is backed by silver and remains strong and sound.